Integral Representations of Fields in Three-Dimensional Problems of Diffraction by Penetrable Bodies
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GRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
Integral Representations of Fields in Three-Dimensional Problems of Diffraction by Penetrable Bodies Yu. A. Eremin1∗ 1
Lomonosov Moscow State University, Moscow, 119991 Russia e-mail: ∗ [email protected]
Received March 12, 2020; revised March 12, 2020; accepted May 14, 2020
Abstract—For the boundary value problem of diffraction of an external disturbance by a local penetrable body with smooth surface, we construct an integral representation of the solution based on a linear combination of single and double layer potentials with densities distributed over a common auxiliary interior surface. A complete mathematical justification of this representation is given. DOI: 10.1134/S0012266120090050
INTRODUCTION The construction of efficient approaches to problems of scattering of acoustic and electromagnetic waves by penetrable bodies continues to attract researchers’ attention. This interest has become stronger owing to the needs of the analysis of scattering properties of thin penetrable scatterers. For practical applications, we can mention diffraction of acoustic waves by protective shields and thin plates [1]; in problems of electromagnetic diffraction, these are ice plates and thin optical lenses [2, 3]. All these applications presume the availability of a reliable numerical analysis technique based on rigorous models that admit efficient computer implementation. The integral equation method [4] has been the basis for constructing models of the kind for a long time. However, as applied to thin transparent bodies, the integral equation method does not always prove to be optimal. For example, the M¨ uller–Kupradze system of Fredholm equations of the second kind often turns out to be degenerate [5, 6]. In this case, one should either use a system of hypersingular equations or construct the solution based on integral representations of fields with densities distributed over auxiliary surfaces [7, 8]. Currently, a similar approach is extensively used in numerous practical applications [2, 9, 10]. In the paper [8], such an approach was applied to solving exterior problems of diffraction by impenetrable bodies. In the present paper, this approach is extended to the case of penetrable scatterers in which one needs to construct a representation of the solution in both exterior and interior domains. It is shown that the solution of such a problem can be constructed based on an integral representation of fields in the form of a combination of single and double layer potentials with densities distributed over a common auxiliary surface localized inside the scatterer. 1. STATEMENT OF THE PROBLEM Consider the problem of diffraction of a given external disturbance—the field u0 —by a penetrable obstacle that is a bounded domain Di in R3 whose boundary is a smooth closed surface ∂Di ∈ C (2,ν) , 0 < ν ≤ 1. The mathematical statement of the boundary value problem has the form 2 4ue,i (M ) + ke,i ue,i (M ) = 0, M ∈ De,i , De = R3 /Di , ∂u0 (Q) ∂ui (Q) ∂ue (Q) − = , Q ∈ ∂Di , (1) ∂n ∂n ∂n ∂ue −
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